The closed, convex hull of every ai c0c0-summing basic sequence fails the FPP for affine nonexpansive mappings

In 2004 Dowling, Lennard and Turett showed that every non-weakly compact, closed, bounded, convex (c.b.c.) subset K   of (c0,‖⋅‖∞)(c0,‖⋅‖∞) is such that there exists a ‖⋅‖∞‖⋅‖∞-nonexpansive mapping T on K that is fixed point free. This mapping T is generally not affine. It is an open question as to whether or not on every non-weakly compact, c.b.c. subset K   of (c0,‖⋅‖∞)(c0,‖⋅‖∞) there exists an affine ‖⋅‖∞‖⋅‖∞-nonexpansive mapping S   that is fixed point free. We prove that if a Banach space contains an asymptotically isometric (ai) c0c0-summing basic sequence (xn)n∈N(xn)n∈N, then the closed convex hull of (xn)n∈N(xn)n∈N, E:=co¯({xn:n∈N}), fails the fixed point property for affine nonexpansive mappings. Moreover, we show that there exists an affine contractive   mapping U:E→EU:E→E that is fixed point free. Furthermore, we prove that for all sequences b→=(bn)n∈N in RR with 0